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@ -95,7 +95,6 @@ First of all (and maybe surprisingly), it is, in most cases, tricky to obtain re
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In the case of vertical excitation energies, \ie, excitation energies at a fixed geometry, band maxima does not usually match theoretical values as one needs to take into account geometric relaxation and zero-point vibrational energy motion.
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Even more problematic, experimental spectra might not be available in gas phase, and, in the worst-case scenario, wrong assignments may have occurred.
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For a more faithful comparison between theory and experiment, although more computationally demanding, the so-called 0-0 energies are definitely a safer playground. \cite{Loo19b}
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%However, they require, from a theoretical point of view, access to the optimised excited-state geometry as well as its harmonic vibration frequencies.
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Another feature that makes excited states particularly fascinating and challenging is that they can be both extremely close in energy from each other and have very different natures ($\pi \ra \pi^*$, $n \ra \pi^*$, charge transfer, double excitation, valence, Rydberg, singlet, triplet, etc).
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Therefore, it would be highly desirable to possess a computational method or protocol providing a balanced treatment of the entire ``spectrum'' of excited states.
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@ -109,41 +108,41 @@ Let us not forget about minimal user input and chemical intuition requirements (
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Finally, low computational scaling with respect to system size and small memory footprint cannot be disregarded.
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Although the simultaneous fulfilment of all these requirements seems elusive, it is essential to keep these criteria in mind.
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%%% TABLE I %%%
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%\begin{squeezetable}
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%\begin{table}
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%\caption{Scaling of various excited state methods.
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%$N$ is the number of basis functions.}
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%\label{tab:method}
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%\begin{ruledtabular}
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%\begin{tabular}{lccc}
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% \mr{2}{*}{Method} & Formal & Oscillator & Analytic \\
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% & scaling & strength & gradient \\
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% \hline
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% CIS & $N^5$ & \cmark & \cmark \\
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% CIS(D) & $N^5$ & \cmark & \cmark \\
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% ADC(2) & $N^5$ & \cmark & \cmark \\
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% CC2 & $N^5$ & \cmark & \cmark \\
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% \\
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% TD-DFT & $N^6$ & \cmark & \cmark \\
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% BSE@GW & $N^6$ & \cmark & \xmark \\
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% ADC(3) & $N^6$ & \cmark & \xmark \\
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% EOM-CCSD & $N^6$ & \cmark & \cmark \\
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% \\
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% CC3 & $N^7$ & \cmark & \xmark \\
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% \\
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% EOM-CCSDT & $N^8$ & \xmark & \xmark \\
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% EOM-CCSDTQ & $N^{10}$ & \xmark & \xmark \\
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% EOM-CCSDTQP & $N^{12}$ & \xmark & \xmark \\
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% \\
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% CASPT2 & $N!$ & \cmark & \cmark \\
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% NEVPT2 & $N!$ & \cmark & \cmark \\
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% FCI & $N!$ & \xmark & \xmark \\
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%\end{tabular}
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%\end{ruledtabular}
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%\end{table}
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%\end{squeezetable}
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\begin{squeezetable}
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\begin{table}
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\caption{Scaling of various excited state methods.
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$N$ is the number of basis functions.}
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\label{tab:method}
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\begin{ruledtabular}
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\begin{tabular}{lccc}
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\mr{2}{*}{Method} & Formal & Oscillator & Analytic \\
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& scaling & strength & gradient \\
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\hline
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TD-DFT & $N^4$ & \cmark & \cmark \\
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BSE@GW & $N^4$ & \cmark & \xmark \\
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\\
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CIS & $N^5$ & \cmark & \cmark \\
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CIS(D) & $N^5$ & \cmark & \cmark \\
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ADC(2) & $N^5$ & \cmark & \cmark \\
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CC2 & $N^5$ & \cmark & \cmark \\
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\\
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ADC(3) & $N^6$ & \cmark & \xmark \\
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EOM-CCSD & $N^6$ & \cmark & \cmark \\
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\\
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CC3 & $N^7$ & \cmark & \xmark \\
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\\
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EOM-CCSDT & $N^8$ & \xmark & \xmark \\
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EOM-CCSDTQ & $N^{10}$ & \xmark & \xmark \\
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EOM-CCSDTQP & $N^{12}$ & \xmark & \xmark \\
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\\
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CASPT2 & $N!$ & \cmark & \cmark \\
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NEVPT2 & $N!$ & \cmark & \cmark \\
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SCI or FCI & $N!$ & \xmark & \xmark \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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\end{squeezetable}
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%**************
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%** HISTORY **%
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@ -216,13 +215,35 @@ Therefore, at the price of a brute force calculation, a SCI calculation is less
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One of the strength of our implementation (based on the CIPSI algorithm developed by Huron, Rancurel, and Malrieu in 1973 \cite{Hur73}) is its parallel efficiency which makes possible to run on thousands of cores.
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Thanks to these tremendous features, SCI methods delivers near FCI quality excitation energies for both singly- and doubly-excited states. \cite{Hol17,Chi18,Loo18a,Loo19c}
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However, although the \textit{``exponential wall''} is pushed back, this type of methods is only applicable to molecules with a small number of heavy atoms with relatively compact basis sets.
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%\cite{Caf16,Gar17b,Hol16,Sha17,Hol17,Chi18,Sce18,Sce18b,Loo18a,Gar18}.
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%%%%%%%%%%%%%%%
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%%% SUMMARY %%%
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%%%%%%%%%%%%%%%
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In summary, each method has its own strengths and weaknesses, and none of them is able to provide accurate and reliable excitation energies for all classes of electronic states, ultimately leading to an unbalanced description of excited states with distinct natures.
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%%% FIG 1 %%%
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\begin{figure*}
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\includegraphics[width=\linewidth]{Set1}
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\caption{Mean absolute error (in eV) with respect to the TBE/aug-cc-pVTZ values from the Jacquemin set \#1 (as described in Ref.~\onlinecite{Loo18a}) for various methods and types of excited states.}
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\label{fig:Set1}
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\end{figure*}
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%%% %%% %%%
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%%% FIG 2 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{Set2}
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\caption{Mean absolute error (in eV) with respect to exFCI excitation energies from the Jacquemin set \#2 (as described in Ref.~\onlinecite{Loo19c}) for various methods taking into account at least triple excitations.}
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\label{fig:Set2}
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\end{figure}
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%%% %%% %%%
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%%% FIG 3 %%%
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\begin{figure*}
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\includegraphics[width=\linewidth]{Set3}
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\caption{Mean absolute error (in eV) with respect to the TBE/aug-cc-pVTZ values from the Jacquemin set \#3 (as described in Ref.~\onlinecite{Loo20}) for various methods and types of excited states.}
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\label{fig:Set3}
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\end{figure*}
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%%% %%% %%%
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%*****************
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%** BENCHMARKS ***
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%*****************
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@ -253,13 +274,13 @@ Finally, let us mention the work of Kannar and Szalay who reported EOM-CCSDT exc
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%%%%%%%%%%%%%%%%%%%%%%%
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Recently, we also made, what we think, is a significant contribution to the quest for highly-accurate excitation energies. \cite{Loo18a}
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More specifically, we studied 18 small molecules with sizes ranging from one to three non-hydrogen atoms.
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For such systems, using a combination of high-order CC, SCI calculations and large diffuse basis sets, we were able to compute a list of 110 highly accurate vertical excitation energies for excited states of various characters (valence, Rydberg, $n \ra \pi^*$, $\pi \ra \pi^*$, singlet, triplet and doubly-excited) based on accurate CC3/aug-cc-pVTZ geometries.
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For such systems, using a combination of high-order CC, SCI calculations and large diffuse basis sets, we were able to compute a list of 110 highly accurate vertical excitation energies for excited states of various characters (valence, Rydberg, $n \ra \pi^*$, $\pi \ra \pi^*$, singlet, triplet and doubly-excited) based on accurate CC3/aug-cc-pVTZ geometries (see Fig.~\ref{fig:Set1}).
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Importantly, it allowed us to benchmark a series of 12 popular excited-state wave function methods accounting for double and triple excitations [CIS(D), ADC(2), CC2, STEOM-CCSD, \cite{Noo97} CCSD, CCSDR(3), \cite{Chr77} and CCSDT-3 \cite{Wat96}].
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Our main conclusion was that, although less accurate than CC3, EOM-CCSDT-3 can be used as a reliable reference for benchmark studies, and that ADC(3) delivers quite large errors for this set of small compounds, with a clear tendency to overcorrect its second-order version ADC(2).
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In a second study, \cite{Loo19c} using a similar combination of theoretical models (but mostly extrapolated SCI energies), we provided accurate reference excitation energies for transitions involving a substantial amount of double excitation using a series of increasingly large diffuse-containing atomic basis sets (up to aug-cc-pVTZ when technically feasible).
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Our set gathers 20 vertical transitions from 14 small- and medium-sized molecules (acrolein, benzene, beryllium atom, butadiene, carbon dimer and trimer, ethylene, formaldehyde, glyoxal, hexatriene, nitrosomethane, nitroxyl, pyrazine, and tetrazine).
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An important addition to this second study was the computation of excitation energies with multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including perturbative and iterative triple corrections.
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An important addition to this second study was the computation of excitation energies with multiconfigurational methods (CASSCF, CASPT2, and NEVPT2) in addition to high-order CC methods including perturbative and iterative triple corrections (see Fig.~\ref{fig:Set2}).
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Our results clearly evidenced that the error in CC methods is intimately related to the amount of double-excitation character in the vertical transition.
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For ``pure'' double excitations (\ie, for transitions which do not mix with single excitations), the error in CC3 and EOM-CCSDT can easily reach $1$ and $0.5$ eV, respectively, while it goes down to a few tenths of an electronvolt for more common transitions (such as in trans-butadiene) involving a significant amount of singles.
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The quality of the excitation energies obtained with multiconfigurational methods was harder to predict as the overall accuracy of these methods is highly dependent on both the system and the selected active space.
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@ -267,7 +288,7 @@ The quality of the excitation energies obtained with multiconfigurational method
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In our latest study, \cite{Loo20} in order to provide more general conclusions, we generated highly-accurate vertical transition energies for larger compounds with a set composed by 27 molecules encompassing from 4 to 6 non-hydrogen atoms for a total of \alert{238} vertical transition energies of various natures.
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To obtain these energies, we used EOM-CC methods up to the highest possible order (CC3, CCSDT, and CCSDTQ), SCI calculations up to several million determinants, as well as the most robust multiconfigurational method, NEVPT2, each approach being combined with diffuse-containing atomic basis sets.
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For all these transitions, we reported at least CC3/aug-cc-pVQZ transition energies and as well as CC3/aug-cc-pVTZ oscillator strengths for all dipole-allowed transitions.
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In agreement with our two previous studies, \cite{Loo18a,Loo19c} we confirmed that CC3 almost systematically delivers transition energies in agreement with higher-level theoretical models ($\pm 0.04$ eV) except for transitions presenting a predominant double excitation character.
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In agreement with our two previous studies, \cite{Loo18a,Loo19c} we confirmed that CC3 almost systematically delivers transition energies in agreement with higher-level theoretical models ($\pm 0.04$ eV) except for transitions presenting a predominant double excitation character (see Fig.~\ref{fig:Set3}).
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%%%%%%%%%%%%%%%%%
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%%% COMPUTERS %%%
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FontSize->Scaled[0.04444444444444444],
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FontColor->RGBColor[
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0.9728288904374106, 0.621644452187053, 0.07336199581899142]], {-118.5, -29.},
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Center, Automatic]},
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AspectRatio->NCache[
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Rational[3, 4], 0.75],
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DefaultBaseStyle->{"Graphics", FontFamily -> "Helvetica"},
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Method->{
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"DefaultBoundaryStyle" -> Automatic, "DefaultPlotStyle" -> Automatic},
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PlotRange->{{-167., 158.}, {-124.5, 122.5}}]], "Output",
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CellChangeTimes->{{3.771316703459606*^9, 3.771316715929142*^9}, {
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3.771316795601022*^9, 3.771316816729157*^9}, {3.771316854360145*^9,
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3.771316948102839*^9}, {3.771317025725174*^9, 3.771317089536953*^9},
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3.781805159269861*^9, {3.781805201492854*^9, 3.7818052078027*^9}, {
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3.781805258995874*^9, 3.781805297562402*^9}, {3.781805363341868*^9,
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3.781805365523806*^9}, {3.781805400588985*^9, 3.781805428943116*^9},
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3.781805515327743*^9, {3.781805663371274*^9, 3.7818056755109463`*^9}},
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CellLabel->"Out[24]=",ExpressionUUID->"9eb27830-1d63-4124-b01c-2f08d28195d1"]
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}, Open ]]
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},
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WindowSize->{808, 655},
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WindowMargins->{{4, Automatic}, {Automatic, 4}},
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FrontEndVersion->"11.3 for Mac OS X x86 (32-bit, 64-bit Kernel) (March 5, \
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2018)",
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StyleDefinitions->"Default.nb"
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]
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(* End of Notebook Content *)
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(* Internal cache information *)
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(*CellTagsOutline
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CellTagsIndex->{}
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*)
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(*CellTagsIndex
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CellTagsIndex->{}
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*)
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(*NotebookFileOutline
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Notebook[{
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Cell[CellGroupData[{
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Cell[580, 22, 2166, 49, 115, "Input",ExpressionUUID->"b227ab8f-bd11-4f6d-9606-4715cb7bdbab"],
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Cell[2749, 73, 4523, 103, 287, "Output",ExpressionUUID->"9eb27830-1d63-4124-b01c-2f08d28195d1"]
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}, Open ]]
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}
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||||
]
|
||||
*)
|
||||
|
||||
Loading…
Reference in New Issue
Block a user